## Abstract

Let [Formula presented], for 0 ≤ b ≤ p^{2} − 1 and gcd(b, p) = 1, be the Fermat quotients arranged in the p × (p − 1) Fermat quotient matrix FQM(p). We study the elements of the matrix and prove two results: (i) Under the Generalized Riemann Hypothesis, ℤ_{p} is fully covered fast by the quotients q_{p}q with q prime and q ≪ plog^{2}p, as p → ∞ and (ii) The matrix FQMp) passes the pair size correlation test, in which any pair (u, v) of entries in the matrix, which lie apart under any apriori chosen fixed geometric pattern, are in the limit, as p → ∞, with equal probability 1/2 in the size relation u ≤ v or u ≥ v.

Original language | English (US) |
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Article number | 112335 |

Journal | Chaos, solitons and fractals |

Volume | 161 |

DOIs | |

State | Published - Aug 2022 |

## Keywords

- Fermat quotient matrix
- Fermat quotients
- Pair correlation size test
- Wieferich primes

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematics(all)
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics